In general, in testing an analog circuit or a mixed-signal circuit, the measurement of analog signals is challenging because analog signals are likely to be affected by noise. Further, the accuracy and stability of sampling in the recipient side also directly influences the test result. From the viewpoint of signal processing, the analysis of the test output can be accelerated if the outputs are digitized signals. Besides, the digitized signals are less likely to be distorted during transmission. Moreover, an external Automatic Test Equipment (ATE) can also process the digitized signals more efficiently.
A digital-analog converter (DAC) is a device that converts a digital, usually binary, signal to an analog signal. One or more DAC modules are commonly integrated in a System-On-a-Chip (SOC). A typical test for a digital to analog converter (DAC) usually includes the following. The analog output terminal of a DAC under test is coupled to the input terminal of a measurement analog to digital converter (ADC). Digital control signals are sent to the input terminal of the DAC. The measurement ADC digitizes the output signals of the DAC and generates digital signals for analysis.
The development in semiconductor technologies has increasingly driven the resolutions and update rates of such DACs higher and higher. For instance, a DAC update rate can be over 1G samples per second (SPS) with resolution as high as 16 bits. This continuously poses challenges for SOC manufacturers for testing high performance DACs with high cost-efficiency and high accuracy.
A known approach to calculate frequency domain parameters from the captured testing data is “FSIR” (Fundamental Signal Identification and Removal) as presented in Akinori Maeda, “Method to Calculate Frequency Domain Parameters of the Non-coherent Waveform”, Verigy Japan, Go/Semi Technical News Letter, March 2011. The entire content of the reference is incorporated by reference herein. This method estimates a fundamental signal through the Tabei-Ueda algorithm and subtracts it from the original waveform in a time domain. Then the residual waveform is converted to frequency domain and the harmonics power and noise power are calculated. However, this approach does not take into account the offset of the harmonic bins, which leads to significant errors in the total harmonic distortion (THD) calculations since the harmonics are treated as noise.
An approach to estimate a non-periodic signal is “FXT” (Fourier Transform Extension) presented in Fang Xu, “Close-in Noise, and Its Application to Converter Test”, IMTC 2006. The entire content of the reference is incorporated by reference herein. This method includes finding the Δphase of the first point and the last point of a captured waveform and removing it by multiplying
      e                  -        j            ⁢                          ⁢      2      ⁢      π      ⁢                        Δ          ⁢                                          ⁢          P                N            ⁢      i        .Then the non-periodic signal is twiddled to a periodic signal without spectral leakages. The FXT method has improved statistic stability in production testing. However, its the application coverage is limited as a Δphase is estimated based on the whole spectrum of a captured signal and may lead to a significant error especially when a phase trend is not monotonic.
In addition, neither the FSIR nor the FXT approaches factors in drift of DC (direct current) offset which may occur in the real-life production environment. A DC drift may adversely affect the spectrum analysis which causes errors in the determination of frequency domain parameters.
Usually, two types of characteristics are tested in the production phase of a DAC module in an integrated circuit (IC) device, e.g., an SOC chip: the static properties and the dynamic properties. The most common static properties to be tested include Differential Non-Linearity (DNL) and Integral Non-Linearity (INL). For dynamic properties, on which embodiments of the present disclosure is focused, SNR (Signal to Noise Ratio) and THD (Total Harmonic Distortion) are most critical properties to be tested.
SNR is defined as a power ratio between the fundamental signal and the background noise. Both the signal and noise power are measured at the same or equivalent points within the same system bandwidth. If the signal and the noise are measured across the same impedance, an SNR can be obtained by calculating the square of the amplitude ratio:
                              S          ⁢                                          ⁢          N          ⁢                                          ⁢          R                =                                            P              signal                                      P              noise                                =                                    (                                                A                  signal                                                  A                  noise                                            )                        2                                              (        1.1        )            where P is the average power and A is the Root-Mean-Square (RMS) amplitude. Due to the wide dynamic range of signals, a logarithmic decibel scale is usually used to represent SNR:
                              S          ⁢                                          ⁢          N          ⁢                                          ⁢                      R            dB                          =                              10            ⁢                                          log                10                            ⁡                              (                                                      P                    signal                                                        P                    noise                                                  )                                              =                      20            ⁢                                          log                10                            ⁡                              (                                                      A                    signal                                                        A                    noise                                                  )                                                                        (        1.2        )            
In such an SNR determination, harmonic distortion components are usually not considered. Equation (1.3) can be used to calculate SNR when an Automatic Test Equipment (ATE) digitizer is used to capture the waveform from a DAC under test.
                              S          ⁢                                          ⁢          N          ⁢                                          ⁢                      R            dB                          =                  20          ⁢                                    log              10                        (                                          A                                  bin                  M                                                                                                                        ∑                                              bin                        =                        1                                                                                              N                          2                                                -                        1                                                              ⁢                                          A                      bin                      2                                                                      ,                                  bin                  ≠                  kM                                ,                                  k                  =                  1                                ,                2                ,                                  3                  ⁢                  …                                                      )                                              (        1.3        )            where M is the fundamental bin number and N is the samples number. Since after a Fast Fourier Transform (FFT), the positive half of spectrum is complex conjugate to the negative half, only the positive half needs to be processed and so the noise bins are calculated from 1 to N/2−1.
Total harmonic distortion (THD) is defined as the ratio of a sum of the powers of all harmonic components to the power of the fundamental frequency.
                              T          ⁢                                          ⁢          H          ⁢                                          ⁢          D                =                                            ∑                              i                =                2                            ∞                        ⁢                          P                                                harmonic                  ⁢                  _                  ⁢                  bin                                ⁢                                  _                  ⁢                  i                                                                          P                          fundamental              ⁢              _              ⁢              bin                                                          (        1.4        )            
In testing, it is impossible to analyze all of the innumerable harmonic components. Commonly a number of harmonics (R) to be included in the THD calculation is predefined. Same as SNR, a logarithmic decibel scale can be used to represent THD, e.g.,
                              T          ⁢                                          ⁢          H          ⁢                                          ⁢                      D            dB                          =                  20          ⁢                                    log              10                        (                                                                                ∑                                          i                      =                      2                                        R                                    ⁢                                                            (                                              A                                                                              harmonic                            ⁢                            _                            ⁢                            bin                                                    ⁢                                                      _                            ⁢                            i                                                                                              )                                        2                                                                              A                                  fundamental                  ⁢                  _                  ⁢                  bin                                                      )                                              (        1.5        )            
Normally frequency domain parameters, like SNR and THD, are most important parameters to evaluate DAC performances. To derive these frequency domain parameters by Fourier Transform, a captured waveform in a time domain needs to have integer number of cycles. This condition is known as coherent sampling.
In production testing, a waveform output from a DAC under test can be captured by an ATE digitizer. Then FFT is applied to derive the frequency domain parameters. FFT is a powerful tool for spectrum analysis and its processing time is fast enough to satisfy the test time requirements of production. On the other hand, the condition of coherent sampling demands an integer number of periods in the waveform within one Unit Time Period (UTP).
An FFT calculation treats a sampled waveform as if it is repeated infinitely. However, if the coherent sampling condition is not met, discontinuity occurs when the waveform is repeated and there will be a large spectral leakage in spectrum analysis. FIG. 1 shows the simulation results of Fourier Transforms 103 and 104 of sinusoidal waveforms 102 and 102 which have 15 and 15.01 periods respectively. As shown, big smearing is present in the spectrum due to a fraction of a period, as shown in 102 and 104.
Unfortunately, due to the hardware and environment limitations, the coherent sampling condition can hardly be satisfied in DAC testing. In past decades, many techniques have been developed trying to solve this issue, such as window functions, time domain interpolation re-sampling and extension of Fourier Transform. Each of these techniques offers unique advantages but has only limited coverage.
Coherent sampling is generally expressed as Equation (1.6):
                                          F            t                                F            s                          =                  M          N                                    (        1.6        )            where Ft is the signal frequency, Fs is the sampling frequency of the digitizer, N is the number of sample points, and M is the periods of the measured waveform in the sampled data (N points). For example, when a sine wave (40 MHz) is measured with 253 periods by 2048 sampled points, the sampling frequency should be 323.794466 . . . MHz. In real-life, the relationship expressed by Equation (1.6) cannot be perfectly satisfied for certain reasons, such as inadequate precision in the frequency setting resolution, a fixed sampling frequency, or an unknown or unstable measured frequency.
The window function techniques can be employed to perform frequency domain analysis without requiring coherent sampling. FIG. 2 shows an example waveform before and after a window function is applied. The thin line 201 shows the original waveform and the bold line shows the processed waveform after the window function is applied (Hanning window). As can be seen, there is no discontinuity of the bold line 202 as the amplitude of waveform declines to 0.
Five types of window functions are commonly used for frequency analysis: Rectangular window, Hanning window, Hamming window, Blackman window and Flat-top window. FIG. 3 shows various window functions in a time domain. FIG. 4 shows the various window functions in a frequency domain.
However, each window function can give good results for only some, but not all, of the frequency domain parameters. For example, the Flat-top window function 301 and 401 results in correct amplitudes of spectrums but large errors in SNR. The Hanning 303 and 403 and the Blackman 302 and 402 window functions result in better SNR results but large errors amplitude determination.